closely connected with the effect of a strong increase in local stresses in a polymer solution flow when the longitudinal velocity gradient reaches the value of reciprocal relaxation time.3
7.2.2THERMAL GROWTH OF BUBBLES IN SUPERHEATED SOLUTIONS OF POLYMERS
Growth of vapor bubbles in a superheated liquid is the central phenomenon in boiling processes. When the bulk superheat is induced by a decrease in pressure, then the initial stage of vapor bubble growth is governed by inertia of the surrounding liquid. During this stage the rheological properties of liquid play important role, discussed in the previous section. The basic features, characterizing this stage, are pressure changes within bubbles and their pulsations. After leveling of pressure in the phases, the process turns into the thermal stage when the cavity growth rate is controlled by ability of the liquid to supply the heat necessary for phase transitions. Expansion of vapor bubble in the thermal regime was examined46 for the case of liquid representing a binary solution. Similar problem was treated47 under additional assumption that the convective heat and mass transfer in the two-component liquid phase is insignificant. More recent works on dynamics of vapor bubbles in binary systems are reviewed elsewhere.48-50
The features, peculiar to vapor bubbles evolution in polymeric solutions at the thermal stage, owe mainly to the following. First, only the low-molecular solvent takes part in phase transitions at the interface because of a large difference in molecular masses of the solvent and polymer. The second, polymeric solutions, as a rule, are essentially non-ideal and, therefore, saturated vapor pressure of the volatile component deviates from the Raul’s law. Finally, the diffusion coefficient in solution is highly concentration dependent that can greatly influence the rate of the solvent transport toward the interface. The role of the listed factors increases at boiling of systems that possess a lower critical solution temperature (LCST) and thus are subjected to phase separation in the temperature range T < Ts, where Ts is the saturation temperature. In the latter case the rich-in-polymer phase which, as a rule, is more dense, accumulates near the heating surface (when a heater is placed at the bottom). As a consequence, the growth of bubbles proceed under limited supply of the volatile component.
Consider the expansion of a vapor cavity in a polymer solution with equilibrium mass concentration of the solvent, k0, at the temperature Tf0 > Ts(k0, pf0), assuming that both pressure and temperature in the vapor phase are constant
Parameters k0, Tf0 characterize the state of solution far from the bubble (at r = ∞). Unlike a one-component liquid, the temperature TfR here is unknown. It is related to the surface concentration of solvent, kR, by the equation of phase equilibrium at the interface.
Equations for heat transfer and diffusion in the solution have the form
∂T
f
+ v
R 2 ∂Tf
= r −2
∂
∂T
f
a
r 2
[7.2.45]
∂t
fR r 2
∂r
∂r
f
∂r
∂k
+ v fR
R 2 ∂k
= r
−2
∂
2
∂k
Dr
[7.2.46]
∂t
r 2 ∂r
∂r
∂r
7.2 Bubbles dynamics and boiling
373
Since the thermal diffusivity of solution, af, is less affected20 by variations of temperature and concentration over the ranges Ts(kR) < Tf < Tf0 and kR < k < k0, respectively, than the binary diffusion coefficient, D, it is assumed henceforward that af = const. Furthermore, since the thermal boundary layer is much thicker than the diffusion layer, it is appropriate to assume that within the latter D = D(k,TfR).
The boundary conditions for equations [7.2.45], [7.2.46] are as
Tf
= Tf0,
k = k0
at
r = ∞
[7.2.47]
&
−1
j,
&
−1
j
[7.2.48]
R − v fR
= ρ f
R
= ρv
&
∂k
∂Tf
j = (R − v fR
)ρ f k R
+ ρ f D
,
jl = k f
at r = R(t )
[7.2.49]
∂r
∂r
where:
j
phase transition rate per unit surface area of a bubble
vfR
radial velocity of the liquid at the interface
kf
heat conductivity of liquid
ρf, ρv
densities of solution and solvent vapor
Equations [7.2.48] and [7.2.49] yield
j =
ρ f
D
∂k
|
r =R
[7.2.50]
1− k R
∂r
If thermodynamic state of the system is far from the critical one, ε = ρ v
/ ρ f << 1 and it
&
&
is possible to assume that vfR = R(1- ε) ≈ R. The solution of equations [7.2.45], [7.2.46] is
searched in the form Tf = T(η),
k = k(η) with η= r/R(t). The concentration dependence of the
diffusion coefficient is represented as D = D0(1 + f(k)). The self-similar solution of the problem exists if
& −1
= const,
& −1
= const
h = RRaf
h1 = RRD0
In this case the functions T(η), k(η) satisfy the following equations:
∂2T
+ [hη + 2η−1 − (1− ε)hη−2 ]
∂T
= 0
∂η2
∂η
∂2k
−1
−1
−1
−2
∂k
−1 df dk
2
+
D
h η + 2η
− D
(1− ε)h η
+ D
= 0, D = D / D
∂η2
]
∂η
[
1
1
dk dη
0
[7.2.51]
[7.2.52]
[7.2.53]
Equation [7.2.53], as opposed to [7.2.52], is non-linear and cannot be solved analytically for arbitrary function D = D(k(η)). Note that in the case of the planar non-linear diffusion, if the self-similarity conditions are satisfied, the problem has analytical solution for particular forms of the dependencies D = D(k) (e.g., linear, exponential, power-law, etc51). However, the resulting relationships are rather cumbersome. The approximate solution of the problem was derived in the case Ja >> 1, using the perturbation method:52
374
Semyon Levitsky, Zinoviy Shulman
h = (6 / π)Ja 2
= (6π)Le −1Di 2(1+ M )2, Di = ε−1K
1+ f (k
) ,
K
= (k
0
− k
R
) / (1− k
R
) [7.2.54]
1
α[
R ]
α
where:
Ja
Jacob number, Ja = cf Tf(εl)-1
Tf
superheat of the solution with respect to the interface,
Tf = Tf0 - TfR
Le
Lewis number, af /D0
Kα
mass fraction of the evaporated liquid46
Here M1 follows certain cumbersome equation,52 including f(k). The approximation Ja>>1 corresponds to the case of a thin thermal boundary layer around the growing bubble. Since, for polymeric solutions Le >> 1, the condition of small thickness of the diffusion boundary layer is satisfied in this situation as well.
We start the analysis of the solution [7.2.54] from the approximation f = 0 that corresponds to D ≈ D0 = const. Then from [7.2.54] it follows:
Kα = (
)c f l −1 T
Le
[7.2.55]
Because of the diffusion resistance, the solvent concentration at the interface is less then in the bulk, kR < k0. Writing the equation of phase equilibrium in linear approximation with respect to k = k0 - k, from [7.2.55] one can receive49,53
= − −1 −
T / T * 1 cf l (1 kR )
∂Ts
−1
T * = Tf 0 − Ts(k0 )
Le
,
[7.2.56]
∂k
k = k
0
Here T* represents the superheat of the solution at infinity. For solutions of polymers
∂Ts/∂k < 0 and, therefore, the actual superheat of the liquid T <
T*. Additional simplifica-
tion can be achieved if 1 - kR >> k0 - kR. It permits to assume in [7.2.56] kR ≈k0 and, hence, to
find easily the vapor temperature.
In the diffusion-equilibrium approximation (i.e. Le → 0)
T = T*. When the diffu-
sion resistance increases, the actual superheat T lowers and, according to [7.2.56], at Le → ∞ T →0. However, in the latter case the assumptions made while deriving [7.2.56], are no longer valid. Indeed, the Ja number, connected with the superheat of the solution with respect to the interface, is related to the Ja0 value, corresponding to the bulk superheat, by Ja=Ja0( T/ T*). Since the ratio T/ T* varies in the range (0, 1), then, at small diffusion coefficients, it may be that Ja << 1 even when Ja0 >> 1. In this case, the asymptotic solution of the problem takes the form46 h = Ja, and, for thin diffusion boundary layer, it can be received instead of [7.2.54]:
h = Ja = (6 / π)Le −1Di 2 (1+ M1 )
[7.2.57]
Finally, at Di << 1 and Ja << 1, the non-linear features in the diffusion transport can be neglected and the expressions for h and kR (or TfR) take the form
h = Ja = Le −1Di
[7.2.58]
The bubble growth in the thermal regime follows the law54
R = Ct, C = 2af h
7.2 Bubbles dynamics and boiling
375
Figure 7.2.10. Limiting superheat at vapor bubble growth in polymeric solution. For all graphs Kρ = 0.7, the symbol “o” corresponds to J²0 = 1. [Reprinted from Z.P. Shulman, and S.P. Levitsky, Int. J. Heat Mass Transfer, 39, 631, Copyright 1996, the reference 52, with permission from Elsevier Science]
Figure 7.2.11. Dependence of the effective Jacob number for a vapor bubble, growing in a superheated aqueous solution of a polymer, on the parameter G. [Reprinted from Z.P. Shulman, and S.P. Levitsky, Int. J. Heat Mass Transfer, 39, 631, Copyright 1996, the reference 52, with permission from Elsevier Science]
where the constant C can be evaluated through h from [7.2.54], [7.2.57] and [7.2.58]. Note that since Ja < Ja0, the bubble growth rate in a polymer solution is always lower than that in a similar one-component liquid.
The set of equations, formulated above, is closed by the equation of phase equilibrium [7.2.37]. The temperature dependence of the pure solvent vapor pressure is described by equation54 p 0v0 = Aexp(-B/T).
Numerical simulations of vapor bubble growth in a superheated solution of polymer were performed,52 using iterative algorithm to account for the diffusion coefficient dependence on concentration in the interval (kR, k0). The results are reproduced in Figures 7.2.10-7.2.12,
where:
Sn
Scriven number, Sn = T/ T*
G
dimensionless parameter, G = εJa0Le1/2
$
superheat of the solution at infinity, evaluated from the condition Sn = 0.99
T *
A characteristic feature of the liquid-vapor phase equilibrium curves for polymeric solutions in the coordinates p, k or T, k is the existence of plateau-like domain in the region of small polymer concentrations (k* ≤k0 ≤1). For this concentration range, the number Ja$ 0 can be defined so that at 1 < Ja0 < J-0the diffusion-induced retardation of the vapor bubble growth does not manifest itself because of weak dependence of Ts (or ps) on kR. The J-0
value or the corresponding limiting superheat,
$
T*, can be estimated from the condition Sn
= 0.99 (i.e. the deviation of effective superheat,
T, from the bulk one, T*, does not exceed
1%). The dependence of the so-defined parameter,
$
T*, on k0 is represented in Figure
7.2.10. For curves 1, 2, 2', 2'': l = 2.3×106 Jkg-1, cf = 3×103 Jkg-1K-1, af = 10-7m2s-1, D0 = 5×10-11m2s-1; for 3, 4: l = 3.6×105 Jkg-1, cf = 2×103 Jkg-1K-1, af = 8×10-8m2s-1, D0 = 5×10-11
Figure 7.2.12. Effect of the solution bulk superheat on the Scriven and Jacob numbers. (−) - solution of polymer in toluene, (- - -) - aqueous solution. [Reprinted from Z.P. Shulman, and S.P. Levitsky, Int. J. Heat Mass Transfer, 39, 631, Copyright 1996, the reference 52, with permission from Elsevier Science]
376 Semyon Levitsky, Zinoviy Shulman
m2s-1; for 1 - 4:
α = 0; for 2', 2'': α = 1, -1; for 1, 3: χ = 0.1; for 2, 4: χ = 0.4. Here
α = kR-1(dD
)
/ dk
, k=k/kR.
k=k 0
$
It is seen that the
T* value decreases with reduction of k0 and/or increasing the
non-linearity factor, α. Raising the value of the Flory-Huggins constant, χ, causes the
$
T*
value to increase and extends the range k* ≤ k0 ≤ 1. The T* value essentially depends on the rate of the diffusion mass transfer; reduction of the latter lowers the limiting superheat, that is the value of T*, below which the bubble grows in a polymeric liquid as though it were a pure solvent. For polymer solutions in volatile organic solvents, the limiting superheat is lower than for aqueous solutions of the same concentrations. Note that for low molecular binary solutions the term “limiting superheat” in the current sense is meaningless in view of pronounced dependence Ts = Ts(k0) in the entire range of the k0 variation. The scale of the effect under consideration is closely connected with the deviation of the solution behavior from the ideal one: the larger is deviation the less is the effect. This can be easily understood, since in the case of a very large difference between molecular masses of the solvent and solved substance, typical for a polymer solution, the graph Ts = Ts(k0), plotted in accordance with the Raul law, nearly coincides with the coordinate axes.20 For this reason, the bubble growth rate in a polymer solution that obeys the Flory-Huggings law, is always lower than in a similar ideal solution.
The reduction of the diffusion mass transfer rate (G ~ (Le)1/2) at a fixed superheat, T*, leads to a substantial decrease in the effective Jackob’s number, Ja. The growth of the content of a polymer in a solution leads to the same result. This follows from Figure 7.2.11 where curves 1 - 5 correspond to k0 = 0.99, 0.95, 0.7, 0.5, 0.3; 2' - 2''': k0 = 0.95; 3' - 3'':
The influence of non-linearity of diffusional transport is higher for diluted solutions. This is explained by a decrease in the deviation of the surface concentration, kR, from the bulk k0 with lowering k0. This takes place due to simultaneous increase in |∂Ts/∂k| that is characteristic of polymeric liquids. The presence of a nearly horizontal domain on the curve Ja = Ja(G) at k0≥0.95 is explained by the existence of the limiting superheat dependent on
the Lewis number.
The role of diffusion-induced retardation increases with the bulk superheat. This reveals itself in reduction of the number Sn with a growth in T* (Figure 7.2.12). For solutions of polymers in volatile organic liquids, such as solvents, the effect is higher than in aqueous solutions. For concentrated solutions the difference between the effective T and bulk T* superheats makes it practically impossible to increase substantially the rate of vapor bubble growth by increasing the bulk superheat. Curves 5 and 5' clearly demonstrate this. They are calculated for solution of polystyrene in toluene at k0 = 0.3, therewith for the curve 5 the dependence of the diffusion coefficient from
7.2 Bubbles dynamics and boiling
377
temperature and concentration D(T,k) was neglected (α = 0), whereas for the curve 5' it was accounted for according to the experimental data.55 Other curves were evaluated with the following parameter values: curves 1 - 5 correspond to α = 0, k0 = 0.99, 0.95, 0.7, 0.5, 0.3; 1 - 4: χ = 0.1. Thermophysical parameters of the liquid and vapor are the same as in the Figure 7.2.10.
Thus, the rate of expansion of vapor bubbles in superheated solution of polymer is lower than in pure solvent due to diffusion resistance. But in diluted solution at rather small superheats the mechanism of diffusional retardation can be suppressed due to a weak dependence of Ts on k0 in this concentration range. Another important conclusion is that in concentrated solutions it is practically impossible to attain values Ja >> 1 by increasing the superheat because of low values of the corresponding Sn numbers.
7.2.3 BOILING OF MACROMOLECULAR LIQUIDS
Experimental investigations of heat transfer at boiling of polymeric liquids cover highly diluted (c = 15 to 500 ppm), low-concentrated (c ~ 1%), and concentrated solutions (c>10%). The data represent diversity of physical mechanisms that reveal themselves in boiling processes. The relative contribution of different physical factors can vary significantly with changes in concentration, temperature, external conditions, etc., even for polymers of the same type and approximately equal molecular mass. For dilute solutions this is clearly demonstrated by the experimentally detected both intensification of heat transfer at nucleate boiling and the opposite effect, viz. a decrease in the heat removal rate in comparison with a pure solvent.
Macroscopic effects at boiling are associated with changes in the intrinsic characteristics of the process (e.g., bubble shape and sizes, nucleation frequency, etc.). Let’s discuss the existing experimental data in more detail.
Figure 7.2.13. Effect of the HEC additives on the boiling curve. 1 - pure water; 2, 3 and 4 - HEC solution with c = 62.5, 125 and 250 ppm, correspondingly. [Reprinted from P. Kotchaphakdee, and M.C. Williams, Int. J. Heat Mass Transfer, 13, 835, Copyright 1970, the reference 52, with permission from Elsevier Science]
One of the first studies on the effect of water-soluble polymeric additives on boiling was reported elsewhere.56 For a plane heating element a significant increase in heat flux at fixed superheat, T = 10-35K, was found in aqueous solutions of PAA Separan NP10 (M = 106), NP20 (M = 2×106), and HEC (M ~ 7×104 to about 105) at concentrations of 65 to 500 ppm (Figure 7.2.13). The experiments were performed at atmospheric pressure; the viscosity of the solutions did not exceed 3.57×10-3Pas. The following specific features of boiling of polymer solution were revealed by visual observations: (i) reduction in the departure diameter of bubbles, (ii) more uniform bub- ble-size distribution, (iii) decrease in the tendency to coalescence between bubbles. The addition of HEC led to faster covering of the heating surface by bubbles during the initial period of boiling and bubbles were
378
Semyon Levitsky, Zinoviy Shulman
Figure 7.2.14. The relation between the relative heat transfer coefficient for boiling PIB solutions in cyclohexane and the Newtonian viscosity of the solutions measured at T=298 K. T = 16.67 K; λ- PIB Vistanex L-100 in cyclohexane, o - PIB Vistanex L-80 in cyclohexane, x - pure cyclohexane. [Reprinted from H.J. Gannett, and M.C. Williams, Int. J. Heat Mass Transfer, 14, 1001, Copyright 1971, the reference 57, with permission from Elsevier Science]
Figure 7.2.15. The average bubble detachment diameter in boiling dilute aqueous solutions of PEO.59 T = 15K. For curves 1-3 the flow velocity v = 0, 5×10-2, and 10-1m/s, respectively. [Adapted, from S.P. Levitsky, and Z.P. Shulman, Bubbles in polymeric liquids,
Technomic Publish. Co., Lancaster, 1995, with permission from Technomic Publishing Co., Inc., copyright 1995]
smaller in size than in water and aqueous solutions of PAA.
Non-monotonous change in the heat transfer coefficient, α, with increasing the concentration of PIB Vistanex L80 (M = 7.2×105) or L100 (M = 1.4×106) in boiling cyclohexane has been reported.57 The results were received in a setup similar to that described earlier.56 It was found that the value of α increases with c in the range 22 ppm < c < 300 ppm and decreases in the range 300 ppm < c < 5150 ppm. Viscosity of the solution, corresponding to αmax value, according to the data57 only slightly exceeds that of the solvent (Figure 7.2.14). Within the entire range of concentrations at supercritical (with respect to pure solvent) superheats, the film boiling regime did not appear up to the maximum attainable value T ~ 60K. The growth of polymer concentration in the region of “delayed ” nucleate boiling led to a considerable decrease in heat transfer.
These findings56,57 were confirmed58 in a study of the nucleate boiling of aqueous solutions of HEC Natrosol 250HR (M = 2×105), 250GR (M = 7×104), and PEO (M ~ (2-4)×106) at forced convection of the liquid in a tube. A decrease in the size of bubbles in the solution and reduction of coalescence intensity were recognized. Similar results were presented also in study,59 where the increase in heat transfer at boiling of aqueous solutions of PEO WSR-301 (M=2×106) and PAA Separan AP-30 (15 ppm < c < 150 ppm) on the surface of a conical heater was observed. In aqueous solutions of PAA with c > 60 ppm the α value began to decrease. With an increase in c the detachment diameter of bubbles decreased (Figure 7.2.15), the nucleation frequency increased, and the tendency to coalescence was suppressed.
Boiling of PEO solutions with c = 0.002 to 1.28% at atmospheric and sub-atmospheric pressures was examined60 for subcoolings in the range 0 to 80K. It was demonstrated that at saturated boiling the dependence of the heat transfer coefficient α on the polymer concentration is non-monotonous: as c grows, α first increases, attaining the maximal value at c≈0.04% , whereas at c = 1.28% the value of α is smaller than in water (α < αs) (Figure 7.2.10). With a decrease in pressure the effect of polymeric additives weakens and for solution with greatest PEO concentration (in the investigated range) the α value increases, approaching αs from below. The critical heat flux densities in PEO solutions are smaller than those for water.
Figure 7.2.16. Heat transfer coefficient for nucleate pool boiling of PEO aqueous solutions. (pf0 = 9.8×103 Pa). Curve 1 corresponds to pure water, for curves 2 - 7, c = 0.01, 0.02, 0.04, 0.08, 0.16 and 1.28%, respectively. [Adapted, from S.P. Levitsky, and Z.P. Shulman, Bubbles in polymeric liquids, Technomic Publish. Co.,
Lancaster, 1995, with permission from Technomic Publishing Co., Inc., copyright 1995]
7.2 Bubbles dynamics and boiling
379
In view of the discussed results, the work61 attracts special attention since it contains data on boiling of dilute solutions, opposite to those reported earlier.56-60The addition of PAA, PEO and HEC to water in concentrations, corresponding to the viscosity increase up to ηp = 1.32×10-3Pas, has brought about reduction in heat transfer. The boiling curve in coordinates q (heat flux) vs. T displaced almost congruently to the region of larger T values with c (Figure 7.2.17a). It was demonstrated61 that the observed decrease in α with addition of polymer to water can be both qualitatively and quantitatively (with the Rohsenow pool boiling correlation for the heat transfer coefficient62) associated with the increase in the solution viscosity (Figure 7.2.17, b). The experiments61 were performed using a thin platinum wire with diameter 0.3 mm.
Explanation of experimental data needs more detailed discussion of physical factors that can reveal themselves in boiling of polymeric solutions. They include possible changes in capillary forces on interfaces in the presence of polymeric additives; absorption of macromolecules on the heating surface; increase in the number of weak points in the solution, which facilitates increase in the number of nuclei; thermodynamic peculiarities of the polymer-solvent system; the effect of macromolecules on the diffusion mass transfer in
evaporation of solvent; hydrodynamics of convective flows in a boiling layer and the motion of bubbles; manifestation of rheological properties of solution.
The capillary effects were indicated as one of the reasons for the intensification of heat transfer, since many polymers (in particular, HEC, PEO, etc.),63 similar to low-molecular surfactants,64 are capable of decreasing the surface tension. As a result, they decrease both the work of the nucleus formation, Wcr, and the critical size of bubble, Rcr:
Figure 7.2.17. Boiling curves for aqueous solutions of PAA Separan AP-30. (a) experimental data; for curves 1 - 6 ηr = 1.00, 1.01, 1.04, 1.08, 1.16 and 1.32, correspondingly; (b) calculations made with the use of the Rohsenow pool boiling correlation; for curves 1 - 5, ηr = 1.00, 1.01, 1.04, 1.16 and 1.32, respectively (ηr = ηp/ηs). [By permission from D.D. Paul, and S.I. Abdel-Khalik, J. Rheol., 27, 59, 1983, reference 61]
380
Semyon Levitsky, Zinoviy Shulman
Wcr = 16 / 3πσ3Φ(θ)[(dp / dT ) T (1− ρv / ρf )]2,
Φ(θ) = 1/ 4(2 + 3cos θ − cos3 θ) [7.2.59]
Rcr = 2σ[(dp / dT) T(1− ρv / ρ f )]−1
where:
θwetting angle
σsurface tension coefficient
However, it should be noted that the integral effect of the heat transfer enhancement, observed in highly diluted solutions, can not be attributed to the capillary phenomena alone, since the main change in σoccurs in the range of low polymer concentrations59 (c < 50 ppm) and further increase in c does not affect the value of σ, whereas the α value continues to grow. PAA, for example, does not behave like surfactants at all. It should be noted also that in the presence of polymer not only the value of σ changes, but also the wetting angle, θ, in the formula [7.2.59]. The latter may lead to manifestation of different behavior.
Absorption of macromolecules onto a heating surface favors the formation of new centers of nucleation. Together with an increase in nucleation sites in the boundary layer of a boiling liquid it explains the general growth in the number of bubbles. Both this factor and reduction in the σvalue for solutions of polymers that possess surface activity, are responsible for a certain decrease in superheat needed for the onset of boiling of dilute solutions.57,60
The decrease in the water vapor pressure due to presence of polymer in solution at c~1% can be neglected. However, if the solution has the LCST, located below the heating wall temperature, the separation into rich-in-polymer and poor-in-polymer phases occurs in the wall boundary layer. At low concentration of macromolecules the first of these exists in a fine-dispersed state that was observed, for example, for PEO solutions.60 The rich-in-poly- mer phase manifests itself in a local buildup of the saturation temperature, which can be significant at high polymer content after separation; in decrease of intensity of both convective heat transfer and motion of bubbles because of the increase in viscosity; and reduction of the bubble growth rate. The so-called “slow” crisis, observed in PEO solutions60 is explained by integral action of these reasons. Similar phenomenon, but less pronounced, was observed also at high enough polymer concentrations.58 It is characterized by plateau on the boiling curves for solutions of PIB in cyclohexane, extending into the range of high superheats.
The main reason for the decrease in heat transfer coefficient at nucleate boiling of polymeric solutions with c ~ 1% is the increase in liquid viscosity, leading to suppression of microconvection and increasing the resistance to the bubbles’ rising. In the presence of LCST, located below the boiling temperature, the role of this factor increases because appearance in the boiling layer of the rich-in-polymer phase in fine-dispersed state. Another
reason for the decrease of α in the discussed concentration range is the decrease in the bub-
ble growth rate at the thermal stage, when the superheat T > T* (Section 7.2.2).
In highly diluted solutions the change in Newtonian viscosity due to polymer is insignificant, and though the correlation between heat transfer enhancement and increase in viscosity has been noticed, it cannot be the reason for observed changes of α. In hydrodynamics, the effect of turbulence suppression by small polymeric additives is known, but it also cannot be considered for such a reason because laminarization of the boundary layer leads to reduction of the intensity of convective heat transfer.65 Nevertheless, the phenomenon of the decrease of hydrodynamic resistance and enhancement of heat transfer in boiling dilute solutions have a common nature. The latter effect was connected
7.2 Bubbles dynamics and boiling
381
Figure 7.2.18. Growth of vapor bubbles on the heating surface at high (a) and low (b) pressures. [Reprinted from S.P. Levitsky, B.M. Khusid and Z.P. Shulman, Int. J. Heat Mass Transfer, 39, 639, Copyright 1996, the reference 66, with permission from Elsevier Science]
with manifestation of elastic properties of the solution at vapor bubble growth on the heating surface.66
The general character of the bubbles evolution at boiling under atmospheric and subatmospheric pressures, respectively, is clarified schematically in Figure 7.2.18. In the first case (at high pressures) the base of a bubble does not “spread”67 but stays at the place of its nucleation. Under such conditions the decrease in the curvature of the bubble surface with time, resulting from the increase in bubble radius, R, leads to liquid displacement from the zone between the lower part of the microbubble and the heating surface. This gives rise to the local shear in a thin layer of a polymer solution. A similar shear flow is developed also in the second case (at low pressures), when a microlayer of liquid is formed under a semi-spherical bubble. As known, at shear of a viscoelastic fluid appear not only tangential but also normal stresses, reflecting accumulation of elastic energy in the strained layer (the Weissenberg effect3). The appearance of these stresses and elastic return of the liquid to the bubble nucleation center is the reason for more early detachment of the bubble from the heating surface, reduction in its size and growth in the nucleation frequency. All this ultimately leads to enhancement of the heat transfer.
The above discussion permits to explain the experimental results.61 Their reasons are associated with substantial differences in the conditions of boiling on a thin wire and a plate or a tube. Steam bubbles growing on a wire have a size commensurable with the wire diameter (the growing bubble enveloped the wire61). This results in sharp reduction of the boundary layer role, the same as the role of the normal stresses. Besides, the bubble growth rate on a wire is smaller than on a plane (for a wire R ~ tn where n < 1/4).67
The elastic properties of the solution are responsible also for stabilization of the spherical shape of bubbles observed in experiments on boiling and cavitation. Finally, the observed reduction in a coalescence tendency and an increase in the bubble sizes uniformity can also be attributed to the effects of normal stresses and longitudinal viscosity in thin films separating the drawing together bubbles.
The linkage between the enhancement of heat transfer at boiling of dilute polymer solutions and the elastic properties of the system is confirmed by the existence of the optimal concentration corresponding to αmax (Figure 7.2.14). Similar optimal concentration was established in addition of polymers to water to suppress turbulence - the phenomenon that also owes its origin to elasticity of macromolecules.1,3,9 Therefore, it is possible to expect that the factors favoring the chain flexibility and increase in the molecular mass, should lead to strengthening of the effect.
The data on boiling of concentrated polymeric solutions20 demonstrate that in such systems thermodynamic, diffusional, and rheological factors are of primary importance.
t
